Pick a point uniformly at random in a square and connect it to the bottom-left corner of the square. Pick a second point uniformly at random in the square and connect it to the top-left corner of the square. What … Continue reading →
is a non-negative random variable such that its expectation is 5 and its variance is 25. What is the maximum probability that ?(Source: quantprof.org) Decompose as , where and are independent, , and with probability , and otherwise. The problem … Continue reading →
If is a discrete random variable taking values in , then the probability-generating function (PGF) of is defined as where is the probability mass function of . PGFs obey all rules of power series with non-negative coefficients. In particular, PGFs … Continue reading →
points are placed uniformly at random on a circle. What’s the probability that all of them lie within a semicircular arc? Label the points . For each , define the event as the event where all points lie on the … Continue reading →
An -sided fair die is rolled repeatedly. What is the expected number of rolls until the total sum becomes at least ? This question can be solved by conditioning on the outcome of the first roll. For , let be … Continue reading →
I recently learned from Allen Downey’s blog that Our World in Data is providing API access to their data. Our World in Data hosts datasets across several important topics, from population and demographic change, poverty and economic development, to human … Continue reading →
The fused lasso, introduced in Tibshirani et al. (2005) (Reference 1), is an extension of the lasso. While the lasso produces sparse models, the fused lasso produces models that are not only sparse, but favor a locally constant coefficient profile. … Continue reading →
Set-up Assume that we have p-values . Assume that they are computed from z-scores (test statistics following normal distributions). Let and let . Without loss of generality, assume that each test statistic has variance 1. With this, we can express … Continue reading →
If are independent variables and is a random vector independent of the ‘s with for all and , it is well-known that also has a distribution. It is also well-known that if and are independent standard normal variables, then has … Continue reading →
The following proposition demonstrates how one can transform a uniform distribution on into a standard Cauchy distribution: Proposition. If and , then has standard Cauchy distribution. Proof: For any , where the second-last equality is due to the fact that … Continue reading →